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Design and Analysis of Algorithms Greedy algorithms, coin changing problem Haidong Xue Summer 2012, at GSU

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What is a greedy algorithm? Greedy algorithm: an algorithm always makes the choice that looks best at the moment Human beings use greedy algorithms a lot – How to maximize your final grade of this class? – How to become a rich man? – How does a casher minimize the number of coins to make a change?

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What is a greedy algorithm? How to maximize your final grade of this class? MaximizeFinalGrade( quizzes and tests ){ if(no quiz and no test) return; DoMyBest(current quiz or test); MaximizeFinalGrade (quizzes and tests – current one); } – This algorithm works very well for students Why is it correct? //Greedy choice

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What is a greedy algorithm? Assuming that your performance of each quiz and test are independent What if you did your best in the current quiz? – You have the chance to get your maximum final grade – The greedy choice is always part of certain optimal solution What if you did not maximize the grades of the rest of the quizzes and tests? – You get a lower final grade – The optimal solution has to contain optimal solutions to subproblems Greedy-choice property! Optimal substructure

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What is a greedy algorithm? To guarantee that a greedy algorithm is correct 2 things have to be proved: – Greedy-choice property: we can assemble a globally optimal solution by making locally greedy(optimal) choices. i.e. The greedy choice is always part of certain optimal solution – Optimal substructure: an optimal solution to the problem contains within it optimal solutions to subproblems. i.e. global optimal solution is constructed from local optimal solutions

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What is a greedy algorithm? How to become a rich man? Problem: maximize the money I have on 8/1/2012 12:00PM A greedy algorithm: Rich(certain time period P){ Collect as much money as I can in the current 3 hours; Rich(P-3 hours); } //Greedy choice

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What is a greedy algorithm? if Rich is implemented by Haydon Rich (between now and 8/1/2012 12pm ) What are the choices I have in the most recent 3 hours? – Finish this lecture like all the other instructors Money collected: 0 – Go to underground, be a beggar, repeatedly say hey generous man, gimme a quarter! Money collected: 2.5 (since k*0.25, k<=10)

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What is a greedy algorithm? What are the choices I have in the most recent 3 hours? – Rob BOA Money collected: 0 (got killed by cops) – Rob my students Money collected: about 300

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What is a greedy algorithm? Which one is the greedy choice? –T–Teach algorithms Money collected: $0 –B–Be a beggar Money collected: $2.5 (since k*0.25, k<=10) –R–Rob BOA Money collected: $0 (got killed by cops) –R–Rob my students Money collected: about $300 //The greedy choice

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What is a greedy algorithm? What happened if I robbed you? – Students Report the criminal immediately Or report it after your final – The instructor Cops confiscate the illicit $300, i.e. -300 Get fired, and lose the stipend of this month, i.e. about -1400 After making this greedy choice, what is the result of Rich

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What is a greedy algorithm? Rich (between now and 8/1/2012 12pm ) Collect as much money as I can in the current 3 hours; Rich (between 1pm today and 8/1/2012 12pm ); } Greedy choice: $300 However there is a influence on the optimal solution to the subproblem, which prevents the instructor from arriving the richest solution : the best of Rich (between 1pm today and 8/1/2012 12pm ) will be around -$1700 Fail to achieve the optimal solution!

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What is a greedy algorithm? Why the greedy Rich algorithm does not work? – After robbing you, I have no chance to be to get the richest solution – i.e. the greedy choice property is violated

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What is a greedy algorithm? How to become a rich man? –T–Teach algorithms Money collected: $0 –B–Be a beggar Money collected: $2.5 (since k*0.25, k<=10) –R–Rob BOA Money collected: $0 (got killed by cops) –R–Rob my students Money collected: about $300 In this problem, we do not have greedy property So, greedy choice does not help And it is very consistent with what you see now Got fired 2.5 -1400 Got killed -infinity +1400

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Coin changing problem An example: – A hot dog and a drink at Costco are $1.50 – Plus tax it is: 1.5*1.08 = $1.62 – Often, we give the cashier 2 $1 notes – She need to give back, 38 cents as change Generally, you never see she gives you 38 pennies. What is algorithm here?

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Coin changing problem Coin changing problem (informal): – Given certain amount of change: n cents – The denominations of coins are: 25, 10, 5, 1 – How to use the fewest coins to make this change? i.e. n = 25a + 10b + 5c + d, what are the a, b, c, and d, minimizing (a+b+c+d) Can you design an algorithm to solve this problem?

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Coin changing problem n = 25a + 10b + 5c + d, what are the a, b, c, and d, minimizing (a+b+c+d) How to do it in brute-force? – At most we use n pennies – Try all the combinations where a<=n, b<=n, c<=n, d<=n – Choose all the combinations that n = 25a + 10b + 5c + d – Choose the combination with smallest (a+b+c+d) How many combinations?

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Coin changing problem What is the recurrence equation? Time complexity?

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Coin changing problem How many sub problems? Time complexity? n If subproblems are solved, how much time to solve a problem?

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Coin changing problem n = 25a + 10b + 5c + d, what are the a, b, c, and d, minimizing (a+b+c+d) How to do it by a greedy algorithm? coinGreedy( n ){ if(n>=25) s = coinGreedy(n-25); s.a++; else if(n>=10) s = coinGreedy(n-10); s.b++; else if(n>=5) s = coinGreedy(n-5); s.c++; else s=(a=0, b=0, c=0, d=n, sum=n); s.sum++; return s; Time complexity? Greedy choice Always choose the possible largest coin It that greedy algorithm correct?

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Coin changing problem Optimal substructure – After the greedy choice, assuming the greedy choice is correct, can we get the optimal solution from sub optimal result? 38 cents Assuming we have to choose 25 Is a quarter + optimal coin(38-25) the optimal solution of 38 cents? Greedy choice property – If we do not choose the largest coin, is there a better solution?

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Coin changing problem For coin denominations of 25, 10, 5, 1 – The greedy choice property is not violated For other coin denominations – May violate it – E.g. 10, 7, 1 – 15 cents How to prove the greedy choice property for denominations 25, 10, 5, 1? – Optimal structure --- easy to prove – Greedy choice property

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Coin changing problem 1. Prove that with coin denominations of 5, 1, it has the greedy choice property Proof: Apply greedy choice: n = 5 + 5c + d In a optimal solution if there is a nickel, the proof is done If there is no nickel: n = d=5 + d Need to prove that: 1+d <= d d=5+d > 1+d For 5, 1, it has greedy choice property, greedy algorithm works

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Coin changing problem 2. Prove that with coin denominations of 10, 5, 1, it has the greedy choice property Proof: Apply greedy choice: n = 10 + 10b + 5c + d – In a optimal solution if there is a dime, the proof is done – If there is no dime : n = 5c + d Since 5c + d>=10 with the conclusion of the previous slide, c>=2 5c + d = 10 + 5(c-2) + d and c+d > 1+c-2+d it cannot be a optimal solution For 10, 5, 1, it has greedy choice property, greedy algorithm works

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Coin changing problem 3. Prove that with coin denominations of 25, 10, 5, 1, it has the greedy choice property Proof: Apply greedy choice: n = 25 + 25a + 10b + 5c + d – In a optimal solution if there is a quarter, the proof is done – If there is no quarter : n = 10b+5c+d Since 10b+5c+d >= 25 if 25 =2, c>=1 – 10b+5c+d = 25 + 10(b-2) + 5(c-1) + d and b+c+d>1+b-2+c-1+d – it cannot be a optimal solution if n>=30, with the conclusion of previous slide, b>=3 – 10b+5c+d = 25 + 10(b-3) + 5(c+1) + d and b+c+d>1+b-3+c+1+d – it cannot be a optimal solution For 25, 10, 5, 1, it has greedy choice property, greedy algorithm works

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